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Existence results for a generalized mean field equation on a closed Riemann surface

Let $Σ$ be a closed Riemann surface, $h$ a positive smooth function on $Σ$, $ρ$ and $α$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -Δu=ρ\left(\dfrac{he^u}{\int_Σhe^u}-\dfrac{1}{\mathrm{Area}\left(Σ\right)}\right)+α\left(u-\fint_Σu\right), \end{align*} where $Δ$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $ρ\in (8kπ, 8(k+1)π)$ for some non-negative integer number $k\in \mathbb{N}$ and $α\notin\mathrm{Spec}\left(-Δ\right)\setminus\set{0}$. Then we obtain existence results for $α<λ_1\left(Σ\right)$ by using the Leray-Schauder degree theory and the minimax method, where $λ_1\left(Σ\right)$ is the first positive eigenvalue for $-Δ$.